1.15.2: Heat Capacities: Solutions: Solutes: Interaction Parameters
- Page ID
- 393964
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)We describe an excess enthalpy \(\mathrm{H}^{\mathrm{E}}\) for a solution prepared using \(1 \mathrm{~kg}\) of water and \(\mathrm{m}_{j}\) moles of solute \(j\) (at fixed \(\mathrm{T}\) and \(\mathrm{p}\)) ion terms of solute-solute enthalpic interaction parameters.
\[\mathrm{H}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\mathrm{h}_{\mathrm{ij}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2} \nonumber \]
The corresponding excess isobaric heat capacity is defined by equation (b).
\[C_{p}^{E}\left(a q ; w_{1}=1 k g\right)=c_{p i j} \,\left(m_{j} / m^{0}\right)^{2} \nonumber \]
where
\[\mathrm{c}_{\mathrm{pij}}=\left(\frac{\partial \mathrm{h}_{\mathrm{ij}}}{\partial \mathrm{T}}\right)_{\mathrm{p}} \nonumber \]
Here \(\mathrm{c}_{\mathrm{pjj}}\) is a pairwise solute-solute interaction isobaric heat capacity [1]. From
\[\mathrm{H}_{1}(\mathrm{aq})=\mathrm{H}_{1}^{*}(\ell)-\mathrm{M}_{1} \, \mathrm{h}_{\mathrm{ij}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2} \nonumber \]
then,
\[\mathrm{C}_{\mathrm{p} 1}(\mathrm{aq})=\mathrm{C}_{\mathrm{pl}^{2}}^{*}(\ell)-\mathrm{M}_{\mathrm{l}} \, \mathrm{c}_{\mathrm{pjj}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2} \nonumber \]
From
\[\mathrm{H}_{\mathrm{j}}(\mathrm{aq})=\mathrm{H}_{\mathrm{j}^{\prime}}^{\infty}(\mathrm{aq})+2 \, \mathrm{h}_{\mathrm{jj}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{m}_{\mathrm{j}} \nonumber \]
then,
\[\mathrm{C}_{\mathrm{pj}}(\mathrm{aq})=\mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{aq})+2 \, \mathrm{c}_{\mathrm{pjj}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{m}_{\mathrm{j}} \nonumber \]
Footnote
[1] For a solution prepared using \(1 \mathrm{~kg}\) of water and \(\mathrm{m}_{j}\) moles of solute (at fixed \(\mathrm{T}\) and \(\mathrm{p}\))
\[\mathrm{C}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{C}_{\mathrm{p} 1}(\mathrm{aq})+\mathrm{m}_{\mathrm{j}} \, \mathrm{C}_{\mathrm{pj}}(\mathrm{aq}) \nonumber \]
Hence
\[\begin{gathered}
\mathrm{C}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \,\left[\mathrm{C}_{\mathrm{pl}}^{*}(\ell)-\mathrm{M}_{1} \, \mathrm{c}_{\mathrm{pjj}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2}\right] \\
+\mathrm{m}_{\mathrm{j}} \,\left[\mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{aq})+2 \, \mathrm{c}_{\mathrm{pjj}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{m}_{\mathrm{j}}\right]
\end{gathered} \nonumber \]
Then,
\[\mathrm{C}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{C}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{m}_{\mathrm{j}} \, \mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{aq})+\mathrm{c}_{\mathrm{pji}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2} \nonumber \]
Since,
\[\begin{gathered}
\mathrm{C}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{C}_{\mathrm{pl} 1}^{*}(\mathrm{aq})+\mathrm{m}_{\mathrm{j}} \, \mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{aq}) \\
\mathrm{C}_{\mathrm{p}}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\mathrm{c}_{\mathrm{pjj}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2}
\end{gathered} \nonumber \]